Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  0ima Structured version   Visualization version   GIF version

Theorem 0ima 5401
 Description: Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
Assertion
Ref Expression
0ima (∅ “ 𝐴) = ∅

Proof of Theorem 0ima
StepHypRef Expression
1 imassrn 5396 . . 3 (∅ “ 𝐴) ⊆ ran ∅
2 rn0 5298 . . 3 ran ∅ = ∅
31, 2sseqtri 3600 . 2 (∅ “ 𝐴) ⊆ ∅
4 0ss 3924 . 2 ∅ ⊆ (∅ “ 𝐴)
53, 4eqssi 3584 1 (∅ “ 𝐴) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ∅c0 3874  ran crn 5039   “ cima 5041 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051 This theorem is referenced by:  csbrn  5514  nghmfval  22336  isnghm  22337  mthmval  30726  0he  37096  0cnf  38762  mbf0  38849
 Copyright terms: Public domain W3C validator